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A330218
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Least BII-number of a set-system with n distinct representatives obtainable by permuting the vertices.
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6
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OFFSET
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1,2
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COMMENTS
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A set-system is a finite set of finite nonempty sets of positive integers.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
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LINKS
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EXAMPLE
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The sequence of set-systems together with their BII-numbers begins:
0: {}
5: {{1},{1,2}}
12: {{1,2},{3}}
180: {{1,2},{1,3},{2,3},{4}}
35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
13: {{1},{1,2},{3}}
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MATHEMATICA
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule, Table[{p[[i]], i}, {i, Length[p]}], {1}])], {p, Permutations[Union@@m]}]];
dv=Table[Length[graprms[bpe/@bpe[n]]], {n, 0, 1000}];
Table[Position[dv, i][[1, 1]]-1, {i, First[Split[Union[dv], #1+1==#2&]]}]
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CROSSREFS
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Positions of first appearances in A330231.
Achiral set-systems are counted by A083323.
BII-numbers of fully chiral set-systems are A330226.
Cf. A000120, A003238, A007716, A016031, A048793, A055621, A070939, A214577, A326031, A326702, A330098, A330101, A330195, A330217, A330229, A330233.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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